Markets with Asymetric Information

Transcription

1 Microeconomics 2 Presentation: Francis Bloch, Slides: Bernard Caillaud Master APE - Paris School of Economics March 6, 2017 (Lecture 9)

2 I. Asymmetric information I.1. Introduction The economy is fundamentally uncertain / risky, not all information is available to the agents when they make their decisions Moreover, agents receive information about the state of nature that may differ from one agent to the other: asymmetric information Firms have better information about their own costs than their rivals or than consumers Citizens have better information about their willingness to pay for a public good than the central planner

3 I.1. Introduction Prominent importance of asymmetric information framework in modern economic theory (IO, Macro, Labor econ., Finance, Public econ.,...) as well as in related social sciences (e.g. Political sciences) Labor market and unemployment: workers know what they are capable of doing, their talent,... employers do not Credit market and credit rationing: entrepreneurs know how risky their projects are, the investors do not Product markets and barriers to entry: firms have better information on their costs than their rivals

4 I.2. Road map for today Up to now, we saw that: Classical market failure easy to fix under perfect information, but much harder when planner has imperfect information, or more precisely when information is asymmetrically distributed between economic agents and the central planner Imperfect (but symmetric) information per se does not affect the efficiency properties of perfectly competitive equilibria provided financial / asset markets are rich enough (complete markets) Today s main message: Asymmetric information in itself is a source of inefficiency, it is the fundamental maket failure.

5 I.2. Road map for today Roadmap: (RE) Equilibrium under asymmetric information may fail to be efficient: Akerlof s (simplified) example Solving the inefficiency problem by transmitting information to the markets before they operate? Hard information transmission (Milgrom): when non-manipulable proofs are available Signaling (Spence): when costly actions can be taken (see Game Theory course) Communication: (Crawford - Sobel): when costless messages can be exchanged (If time permits...)

6 II. Akerlof s market for lemons Example of a market under asymmetric information N sellers, selling one unit of the good each Units of good can be of quality q {L, H} There are N H = λn sellers of H-quality and N L = (1 λ)n sellers of L-quality A seller of quality q gets utility r q m if she keeps her good if she sells her good at price m M N buyers, interested in buying one unit each A buyer buying a good of quality q at price m gets utility v q m and he gets 0 if he does not acquire the good.

7 II. Akerlof s market for lemons quality % buyer s value seller s value High λ v H = 50 r H = 40 Low 1 λ v L = 30 r L = 25 Efficient allocation: all goods of all qualities transferred from sellers to buyers Perfect information: q observable, 2 markets prices p q Competitive prices: (p L = v L, p H = v H ) since v q > r q ; as expected, it is efficient Imperfect but symmetric information: q unobservable (by sellers and by buyers): one market with price p Competitive price: p = v e = λv H + (1 λ)v L since p = v e r e = λr H + (1 λ)r L ; equilibrium is efficient

8 II. Akerlof s market for lemons p q = v q r q N q M

9 II. Akerlof s market for lemons quality % buyer s value seller s value High λ v H = 50 r H = 40 Low 1 λ v L = 20 r L = 25 Efficient allocation: only H-quality should be traded Imperfect symmetric information: p = v e is a competitive equilibrium iff v e r e, and then all goods are traded: ex-post inefficient equilibrium...! But, it is constrained efficient: i.e. optimal given the information publicly available in the economy Nobody knows q; as if good is of homogenous (average) quality, with buyers value v e and sellers value r e.

10 II. Akerlof s market for lemons quality % buyer s value seller s value High λ v H = 50 r H = 40 Low 1 λ v L = 30 r L = 25 Under asymmetric information: only sellers know the quality of their good q. Sellers competitive supply curve (they re under full information): 0, if p < r L N L = (1 λ)n, i.e. L-quality goods, if r L p < r H N, i.e. all goods, if r H p.

11 II. Akerlof s market for lemons quality % buyer s value seller s value High λ v H = 50 r H = 40 Low 1 λ v L = 30 r L = 25 REE: buyers should extract information from the price, assuming sellers are rational, to formulate their demand Buyers willing to buy if E [v q p] p. Hence their demand: For r L p < r H, E [v q p] = v L, D(p) = M if p v L For r H p, E [v q p] = v e, D(p) = M if p v e For p < r L, no supply; assume e.g. beliefs E [v q p] = v e Demand is not monotonic (if v e > r H ): we can expect problems!

12 II. Akerlof s market for lemons v H v e r H v L r L M

13 II. Akerlof s market for lemons quality % buyer s value seller s value High λ v H = 50 r H = 40 Low 1 λ v L = 30 r L = 25 Suppose λ 1/2: there are many H-quality sellers. Price p = v e r H clears the market: all goods traded This RE Equilibrium is efficient But price p with: r L p = v L r H also clears the market: only L-quality is traded. This L-quality RE Equilibrium is inefficient! Two competitive equilibria that are Pareto ranked.

14 II. Akerlof s market for lemons v H v e r H v L r L N L N M

15 II. Akerlof s market for lemons quality % buyer s value seller s value High λ v H = 50 r H = 40 Low 1 λ v L = 30 r L = 25 Suppose now λ < 1/2: there are few H-quality sellers. There cannot exist an equilibrium with all goods traded since r H = 40 > v e : no price is high enough for H-quality sellers to sell and low enough for buyers to buy the average quality There is adverse selection: L-quality goods kick H-quality goods out of the market. At moderate price, only L-quality sellers are all willing to sell, they drive expected quality down, hence also the price, which prevents H-quality sellers to sell... p = v L is now the unique equilibrium and it is inefficient

16 II. Akerlof s market for lemons v H r H v e v L r L N L N M

17 II. Akerlof s market for lemons Adverse selection in the labor market: Heterogeneity of workers with respect to their talent, related to their productivites; private information Higher talent workers also have better alternative occupations (i.e. work as an independent) For wage ω equal to average productivity in population, labor supply only by workers with talent below a threshold corresponding to the alternative that is equivalent to ω Hence, for ω, the average productivity of workers participating in the labor market smaller than average productivity, there is no demand for labor at ω Lowering ω kicks medium productivity workers out of the market, hence reduces average productivity further...

18 II. Akerlof s market for lemons Conclusion: Asymmetric information may lead to competitive (RE) equilibria that are inefficient: failure of the st Welfare Theorem. The market may even completely collapse: non-existence of non-degenerate (active trade) equilibrium The strength of the competitive equilibrium paradigm rests on existence and efficiency results: less appeal under asymmetric information! Two routes from here: 1 Circumvent the asymmetry of information by having information transmitted to the market (rest of today) 2 Taking asymmetry as given, provide a more appropriate model of elementary transactions under asymmetric information (Principal - Agent models)

19 III. Transmission of information - III.1. Introduction In the Akerlof s example, sellers of high-quality goods would like to convince buyers that they sell high quality Yet, sellers of low-quality would be willing to fool the market and pretend they sell high quality as well! Hence, information transmission has to be made convincingly, in a credible way: By disclosing of hard information (i.e. non-manipulable proof of information) By undertaking observable and costly actions that signal information By communicating information (costless actions)

20 III.1. Transmission of information Formally, this means introducing a pre-market stage, where informed agents make decisions (disclose proofs, emit signals, communicate) Two stages: pre-market actions and then (competitive) market mechanism (hence, not strictly-speaking a game). Information may be revealed: directly: hard proof, message indirectly, as a (bayesian) strategy maps an agent s information into the set of his possible actions: inverting the mapping yields information Typical questions in such a setting: full / partial / no revelation of information? Efficiency?

21 III.2. Disclosure of certifiable / hard information Introduce in Akerlof s example a technology that allows a seller to certify / prove the quality of his good at a cost K before the market opens If seller certifies his quality q, he obtains price equal to v q So high-quality sellers have an incentive to certify provided K is not too large, as otherwise they do not sell (λ small) or they sell at price v e < v h (λ large) Low-quality sellers don t Immediate equilibrium: high-quality sellers show proofs, hence absence of proof induces beliefs that quality is low Full equilibrium disclosure of information to the market

22 III.2. Disclosure of certifiable / hard information Disclosure of hard information (Milgrom) In models with richer information (more than 2 types), there is full disclosure of hard information if the cost of disclosure K is small enough. About efficiency: Certification may improve efficiency when K negligeable and λ < 1/2 and then the right amount of information is disclosed to the market It may be wasteful when there exists an efficient equilibrium without certification, as the cost K is wasted. In general, too much disclosure of information: disclosure is decided on the basis of the private value of information for informed agents, not on the basis of the social value of information

23 III.3. Signaling in Akerlof s model Coming back to the example of the labor market: it is probably difficult for a worker to provide a proof of his/her analytical skills (as determining productivity in some type of job) But people do different things: e.g. they invest in education! Education may of course increase your analytical skills, hence your productivity (APE certainly does...!) But education is also a way to send information about how intrinsically good you are! Passing micro 2 speaks about your intrinsic analytical skills It is harder for one who has low analytical skills to achieve a good grade than for one who has high analytical skills: observing a good grade may be a signal of high intrinsic analytical skills.

24 III.3. Signaling in Akerlof s model quality % buyer s value seller s value high λ v H = 50 r H = 40 low 1 λ v L = 30 r L = 25 Specific Akerlof example with a performance test: Each seller can run a test and achieve a publicly observable level of performance g 0; the test has no impact on the intrinsic quality q but the cost of achieving performance level g varries with quality: C(g, q) = g if q = L = αg if q = H with α < 1 2. Goods differentiated by disclosed performance: p(g) and buyers draw inferences when observing a performance level

25 III.3. Signaling in Akerlof s model quality % buyer s value seller s value high λ v H = 50 r H = 40 low 1 λ v L = 30 r L = 25 An equilibrium is: (g L, g H, p(.), µ(.)), where g q is the level of performance chosen by sellers of quality q, µ(g) are buyers (common) beliefs about quality when observing performance level g R + and p(g) is the equilibrium price for goods with label g, such that: p(g) = µ(g)v H + (1 µ(g))v L (competition among buyers given beliefs) q-type sellers are better off choosing g q rather than deviating and choosing another g Buyers beliefs are consistent with Bayes law, whenever this has some bite

26 III.3.a. Separating equilibrium in Akerlof s model quality % buyer s value seller s value high λ v H = 50 r H = 40 low 1 λ v L = 30 r L = 25 Separating equilibrium g H g L, p(g), µ(g) beliefs after any g Buyers beliefs: µ(g H ) = 1, µ(g L ) = 0 and so competition induces p(g H ) = v H = 50, p(g L ) = v L = 30 Take worst possible beliefs for all other g: g g h, µ(g) = 0 and p(g) = v L = 30 q L -sellers: g L rather than g H, another g or not selling: p(g L ) g L sup{p(g H ) g H, p(g) g, r L } NC: g L = 0 and g H q H -sellers: g H rather than g L, another g or not selling: p(g H ) αg H sup{p(g L ) αg L, p(g) αg, r H } NC: 50 αg H 40

27 III.3.a. Separating equilibrium in Akerlof s model quality % buyer s value seller s value high λ v H = 50 r H = 40 low 1 λ v L = 30 r L = 25 Continuum of separating equilibria: Fix any y with 20 y 10 α, then g H = y, g L = 0, µ(y) = 1, p(y) = 50, g y, µ(g) = 0, p(g) = 30 is a separating equilibrium. More convincing (monotonic) beliefs: µ(g) = 0, p(g) = 30 for 0 g < y, and µ(g) = 1, p(g) = 50 for y g Least-cost separating equilibrium: g H = 20, minimal level of performance that enables q H -sellers to signal themselves

28 III.3.a. Separating equilibrium in Akerlof s model p p-g=cst p h =50 p- g=cst r h =40 p l =30 r l =25 g=0 g=20 g

29 III.3.b. Non-separating equilibrium in Akerlof s model quality % buyer s value seller s value high λ v H = 50 r H = 40 low 1 λ v L = 30 r L = 25 Non-revealing equilibrium g H = g L = g, p(g), µ(g) after any g Buyers don t get any information: µ(g ) = λ, so competition leads to p(g ) = v e For all other g g, take worst beliefs µ(g) = 0 and so p(g) = 30 q L -sellers: g rather than any other g or not selling: p(g ) g sup{p(g) g, r L } So, v e g 30 > 25 = r L q H -sellers: g rather than any other g or not selling: So, v e αg 40 p(g ) αg sup{p(g) αg, r H }

30 III.3.b. Non-separating equilibrium in Akerlof s model quality % buyer s value seller s value high λ v H = 50 r H = 40 low 1 λ v L = 30 r L = 25 Continuum of pooling equilibria: Fix any z such that 0 z inf{v e 30, ve 40 α }, then g H = g L = z, µ(z) = λ, p(z) = v e and g z, µ(g) = 0 and p(g) = 30 is a pooling equilibrium (Remark: As v e > r h = 40, inf{v e 30, ve 40 α } > 0) More reasonable beliefs such as: µ(g) = 0, p(g) = 30 for 0 g < z, and µ(g) = λ, p(g) = v e for z g No information at all is revealed in equilibrium

31 III.3.b. Non-separating equilibrium in Akerlof s model p p-g=cst p h =50 v e r h =40 p- g=cst p l =30 r l =25 g=0 g=20 g

32 III.3.c. Efficiency of signaling in Akerlof s model Natural idea: in separating signaling equilibrium, decentralized information passed onto the market, which restores efficiency Not so simple: the use of signals is driven by the private value of the signal for sellers, not by its social value If λ < 1/2: trade is inefficient (only q L traded) without test technology but efficient under signaling: great! but the cost of test g H > 0 is a pure social waste... here, overall more efficient with than without test technology If λ > 1/2: trade may be efficient (in one equilibrium) with (in separating and pooling equilibria) and without test technology; but social cost of signals.

33 IV. Cheap talk models, costless signals To transmit information about their talent, people also talk: labor market interviews... They can send very positive messages about their productivity: in contrast to education, which involves a cost in acquiring diploma and obtaining good grades that depends upon intrisinc talent, it is not less costly to say that one is extremely good when one actually is rather than when one is not: no differential cost of messages, talk is cheap So, can information be actually transmitted through costless communication? And if so, how much?

34 IV. Cheap talk models, costless signals Elementary Sender-Receiver game that is now the basis of all papers dealing with communication Utility: Information: S informed about state of nature θ [0, 1], while R only has a prior F (.) Moves: S sends a costless (cheap talk) message m [0, 1] to R, then R takes a decision y R U R (y, θ) = (y θ) 2, maximum for y R (θ) = θ U S (y, θ) = (y θ b) 2, maximum for y S (θ) = θ + b b measures the lack of congruence between S and R s objectives

35 IV. Cheap talk models, costless signals Concentrate on (m.s.) Perfect Bayesian Equilibria: y(m), decision rule after any message m, q(. θ), distribution over messages conditional on any θ r(. m), R s beliefs on the state of nature after any message m, Say that θ 0 induces action y 0 if: {m;y(m)=y 0 } q(m θ 0)dm > 0

36 IV. Cheap talk models, costless signals If R does not listen to S, S has no reason to send meaningful messages, and if S sends meaningless messages, R has no reason to listen to S Babbling equilibria There always exists a babbling equilibrium in cheap talk games; here q uniform over [0, 1] for all θ, y(θ) = E[θ] and r = f. Question: Are there other PBE where some information is revealed, and if so, how much information is revealed? Central result: Full information transmission is impossible!

37 IV. Cheap talk models, costless signals Central lemma: a finite number of actions induced If y and y are induced actions in equilibrium, then: y y b; hence, the set of induced actions is finite. Proof: take y < y U S (y, θ ) U S (y, θ ) and U S (y, θ ) U S (y, θ ) So, there exists θ with U S (y, θ ) = U S (y, θ ) and y < θ + b = y +y 2 < y y cannot be induced by θ > θ, so the support of beliefs leading to y must be included in [0, θ ] and so, y θ. Therefore: y θ + b > θ y, i.e. y y > b

38 IV. Cheap talk models, costless signals Important result: even locally, perfect revelation of information is impossible as soon as objectives are not perfectly aligned (b > 0) One can characterize partition PB equilibria, where [0, 1] is split into p intervals, and in each interval, S sends the same message: i.e. there is partial / coarse information revelation Moreover, there exists an upper bound P (b) such that, only partition equilibria with p P (b) exist P (b) decreasing in b: more congruent objectives may lead to more information being communicated / conflict limits communication

39 V. Conclusion Asymmetric information is a major impediment to markets functioning properly It makes classical market failures more difficult to solve And it is the source of inefficiency by itself There are no easy way to fill the informational gap by sending information convincingly to the market, without generating residual inefficiencies Need for a re-formalization of transactions under asymmetric information to provide better foundations to the analysis of markets under asymmetric information

40 Required Reading * Akerlof, G. (1970), Quarterly Journal of Economics, 89, Cho, I. and D. Kreps (1987), Quarterly Journ. 102(2), of Econ., Crawford, V. and J.Sobel (1982), Econometrica, 50(6), Grossman, S. (1981), Journ. Law and Econ., 24, * MC - W - G, Ch 13 Mailath, G. (1987), Econometrica, 55, Milgrom, P. (1981), Bell Journal of Economics, Vol 12 (2), * Spence, M. (1973), Quarterly Journal of Economics, 87,